[SOLVED] metric space 2

[dwsmith]

Consider the following two metric spaces in $\mathbb{R}^n$:
$$
d_2(\mathbf{x},\mathbf{y}) = \sum_{n = 1}^{n}|x_i - y_i|.
$$
Prove that the ball $B(\mathbf{a},r)$ has the geometric appearance indicated:


In $(\mathbb{R}^2,d_2)$, a square with diagonals parallel to the axes.

Take a ball $B((x,y),r)$ for $r > 0$. Let $(a,b)\in B((x,y),r)$.
By the metric, we have $||x-a| + |y-b|| < r$
Now what?

 

[Evgeny.Makarov]

Take a ball $B((x,y),r)$ for $r > 0$. Let $(a,b)\in B((x,y),r)$.
By the metric, we have $||x-a| + |y-b|| < r$
Now what?

As usual with absolute values, consider several cases that allow dropping the absolute value. In this situation, there are four cases:

(1) x < a, y < b;
(2) x < a, y ≥ b;
(3) x ≥ a, y < b;
(4) x ≥ a, y ≥ b.

For example, when (3) holds, $|x-a| + |y-b| < r$ turns into

$$x - a + b - y < r$$

or

$$y > x + (b - a - r)$$